The present invention relates to a discrete pattern. More particularly, the present invention relates to a discrete pattern having low discrepancy and including a pattern of dots arranged so there is no overlapping, an optical member, a light guide plate, a side light device and a light-transmitting liquid crystal display device that uses the discrete pattern, and to a method and a program for generating the discrete pattern, a computer-readable storage medium on which a computer-readable program is stored for generating the discrete pattern, and a discrete pattern generation system.
Various techniques using discrete pattern are known. These techniques are used for a light guide plate, for example, of light-transmitting liquid crystal display device or a diffuser sheet, and the discrete pattern can be, for example, a dithering pattern, a lithography photomask pattern or a pattern for stopper. Recently, the application of a discrete pattern for a DNA arrangement on a DNA chip has also been discussed.
Conventionally, a discrete pattern is formed so that dots are arranged at random by using a so-called random-number generator, or the dots are arranged on a regular orthogonal lattice, such as plotting paper. However, with the conventional methods, the following problems have arisen for the random pattern that is generated merely by a common random-number generator. Specifically, even when dots are arranged at random, the overlapping of dots and uneven dot densities, which occur because each dot has a definite size, can adversely affect the appearance of a dot pattern, and an optical malfunction, such as uneven luminance, can occur. Further, when the dots are regularly arranged, an undesirable optical pattern, such as moire, can occur due to an interference between dots or with an external regular pattern.
To resolve the problem, a method is proposed in Japanese Unexamined Patent Publication No. Hei 10-153779 whereby an irregular pattern is generated without an excessive approach between the dots. According to this method, first, as an xe2x80x9cabsolute random number arrangement methodxe2x80x9d, (1) the initial position (x, y) is provided for all the dots by a random-number generator; and (2) a random number is again generated for the overlapping dots and their positions are corrected. However, as for the calculation method disclosed in Japanese Unexamined Patent Publication No. Hei 10-153779, it is well known that the calculation performed to eliminate the overlapping of dots can not be convergent in an area having a dot filling rate of more than 50%. Specifically, when the method described in Japanese Unexamined Patent Publication No. Hei 10-153779 is used, it is quite difficult for an irregular pattern having no abnormal approach between dots to be generated, while maintaining randomness. Further, according to this method, which is based on the generation of multiple pseudo random numbers, it is also difficult to remove an uneven portion from a dot pattern, even if overlapping of dots can be removed from an area having a low dot filling rate.
FIG. 1 is a diagram showing an example dot pattern that is formed by the method described in Japanese Unexamined Patent Publication No. Hei 10-153779. The dot pattern in FIG. 1 is generated using a process whereby (1) dots are arranged at two-dimensional regular lattice points formed by linear lines or curves, and these positions are defined as initial positions; (2) the displacement of each initial position is provided by a random-number generator; and (3) a random number is again generated for the overlapping dots and their positions are corrected. As is shown in FIG. 1, according to this method, the dots can be arranged without overlapping, so long as only small displacements of the lattice points are maintained. However, according to a method for generating a random position as a perturbation from a lattice point, in an area having a comparatively high dot filling rate, such as a ratio exceeding 50%, it is difficult to generate a satisfactory irregular pattern while avoiding the occurrence of moire. Further, according to this method, which is based on the multiplex generation of pseudo random numbers, many aggregations of dots appear even if dot overlapping is removed, and it is difficult to generate a uniform random pattern.
The reasons for this will be explained. For the conventional example using random dots, assume that the size of the dots being arranged is about 100 xcexcm and that the filling rate is 70%. As is shown in FIG. 1, the shape of a dot is assumed to be a square. With this dot size and the filling rate described above, an interval of merely 20 xcexcm is generated between the dots. In FIG. 1, dot 100 and intervals 102 between the dot 100 are shown at an exact reduced scale. When random perturbation is provided for the regular lattice, dots 104, indicated by broken lines in FIG. 1, are obtained. It is understood that these dot 104 can only generate a pattern having an extremely limited irregularity (hereinafter referred to as randomness in this invention), as is shown in FIG. 1. Because the adjacent dots do not jump over each other, and because the filling rate is high, the positions of these dots are corrected only within a limited range.
A square lattice is employed in the conventional art in FIG. 1. However, under a condition wherein dots should not be arranged too closely together, the randomness of the dot pattern is more or less limited, even for other types of regular lattices. That is, according to the method whereby a predetermined dot is provided at an initial position and perturbation is employed to generate a random arrangement, in principle, as the filling rate increases, the possibility that a random dot pattern will be obtained that closely resembles a truly random arrangement decreases. Therefore, regardless of the filling rate, this method is not satisfactory for the generation of random patterns.
In addition, relative to the optical characteristic of a dot pattern generated by the above method, another problem has arisen in that a moire pattern will occur when a light beam is transmitted through or reflected from a dot pattern. Conventionally, in the printing field, many studies and proposals have been made to devise methods for removing moire. For example, in Japanese Unexamined Patent Publication No. 2000-94756, for a halftone process performed by a printer, a printing technique is disclosed for avoiding the occurrence of moire (so-called uneven feeding and uneven lines) caused by regular printing fluctuations, produced by the rotation of a paper feed drum, and printing dot patterns.
Therefore, in Japanese Unexamined Patent Publication No. 2000-94756, printing dots are arranged at random. That is, perturbation is provided at random for printing dots arranged on a regular lattice, an improvement that satisfactorily precludes the occurrence of the uneven dot densities that accompany the appearance of moire. However, for the above mentioned reasons, it is difficult for the randomness of this method to be directly applied for uniform discrete patterns.
The above problems, including the generation of random dot patterns and the occurrence and removal of moire, arise not only in the printing field, wherein printing techniques for printers are affected, but also in various other fields, such as the production of display devices that include rear lighting devices (hereinafter referred to as backlights).
As a specific example, since light, compact light-transmitting liquid crystal display devices can be made that consume little power, the production and use of these display devices have become ever more important as a core technique affecting the selection of hardware for use in the so-called IT revolution. However, since unlike conventional display devices, i.e., CRTs, liquid crystal devices do not emit light, backlight units must be provided that light all the liquid crystal cells in these devices. This is especially true today, since there is an increased demand for liquid crystal display devices having high resolution color displays, so that accordingly, a fervently desired characteristic of backlight units is that they uniformly, and brightly, illuminate large areas.
FIG. 2 is a diagram showing a light-transmitting liquid crystal display device, a typical display device that includes a backlight unit. An explanation will now be given, using the light-transmitting liquid crystal display device in FIG. 2, for a countermeasure for random dot patterns and moire. As is shown in FIG. 2, a conventional light-transmitting liquid crystal display device includes a backlight unit. The backlight unit includes: a light guide plate 108, on which a random dot pattern 106 is formed; a fluorescent tube CFL, which is arranged adjacent to the light guide plate 108; a reflector 112, which covers the fluorescent tube CFL to ensure efficient transmission, to the light guide plate 108, of light emitted by the fluorescent tube CFL; and a reflection sheet 114, which is used to efficiently reflect, to a liquid crystal panel (not shown), light scattered by the light guide plate 108.
Since, to the extent possible, the dot pattern 106 on the light guide plate 108 is formed at random, problems such as those presented by moire are resolved. As is shown in FIG. 2, a diffusion sheet 116 and prism sheets 118a and 118b, provided for the backlight unit, regulate the distribution of the light irradiating the liquid crystal panel.
The side light type backlight unit shown in FIG. 2 is frequently employed for small devices, such as notebook computers. In the backlight unit in FIG. 2, light emitted by a cold cathode fluorescence light (CFL), such as a fluorescent tube, is scattered by the dot pattern 106 formed on the bottom of the light guide plate 108, which is made of an acrylic resin, or the reflection sheet 114, which is arranged below the light guide plate 108. The thus scattered light is then transmitted to the liquid crystal panel (not shown), passing en route through the diffusion sheet 116 and the prism sheets 118a and 118b, which are arranged above the top face of the light guide plate 108. The thus distributed light can then be viewed by a user. That is, the backlight unit shown in FIG. 2 is a device for converting a line light source into a flat light source.
So long as the so-called side light device shown in FIG. 2 is employed, a mechanism for scattering light emitted by a light source and for reflecting the scattered light onto a liquid crystal panel is indispensable, and is an important device mechanism for improving the luminance produced by the backlight unit. Therefore, various studies have been initiated to examine manufacturing processes employed for the bottom of the light guide plate 108 or for the reflection sheet 114. For example, in Japanese Unexamined Patent Publication No. Hei 8-085001, the bottom of a light guide plate is processed by a cutting tool having a negative rake angle, so that the resultant irregular surface works as the light scattering plane. However, according to this method, it is difficult to quantitatively control the uniformity of the luminance provided by a backlight unit, and since scattered light is wasted due to its deflection in light scattering directions, the method is not appropriate for obtaining a high quality backlight unit that provides a high luminance.
Other methods have also been proposed: a method in Japanese Unexamined Patent Publication No. Hei 7-294745, whereby a groove having a recessed portion in cross section is formed in the bottom of a light guide plate and for scattering light on the top face of the light guide plate; and a method in Japanese Unexamined Patent Publication No. Hei 6-242320, whereby a pattern coated with a particle pigment, such as titanium dioxide, is formed on the bottom of a light guide plate. The above conventional methods include the same feature that provides for the formation of a light scattering structure having a specific geometrical, cyclic design, i.e., a dot pattern, is formed on a light guide plate. However, since generally an element including a small cyclic pattern, such as a color filter or a prism sheet, is indispensable for a liquid crystal display device, when the arrangement of a dot structure is cyclic, the dot structure and the light optically interfere with each other and generate a moire pattern. Since this moire pattern drastically deteriorates the value of a luminous object as a light source, it is preferable that, to the extent possible, the occurrence of moire be avoided.
Relative to the moire pattern described above that occurs as a result of a dot pattern formed on the light guide plate 108 of a display device, such as a light-transmitting liquid crystal display device using a backlight unit, the reduction of moire, as it is related to the various device techniques described above, has also been discussed.
For example, in Japanese Unexamined Patent Publication No. Hei 9-269489, a method for scattering light is disclosed whereby multiple, small light-scattering members, such as micro dots, are arranged at random on the bottom of a light guide plate. Further, an improved method is disclosed in Japanese Unexamined Patent Publication No. 2000-171797, and a method is disclosed in Japanese Unexamined Patent Publication No. Hei 11-250713 whereby dots are arranged at random on the top of a light guide plate in order to employ them for a reflection type liquid crystal display device. FIG. 3 is a diagram showing a conventional example for which a dot pattern generated at random is employed for a reflection type liquid crystal display device.
In the conventional case in FIG. 3, a light guide plate on which a random dot pattern is formed using a pseudo random number is used to constitute a backlight unit. In the backlight unit in FIG. 3, a fluorescent tube CFL and a reflector 124 are arranged adjacent to a light guide plate 122 on which multiple dots 120 are formed at random using a pseudo random number. The light guide plate 120, the fluorescent tube CFL and the reflector 124 in FIG. 3 are supported by a frame 126 and together constitute a backlight unit that reflects light in the direction indicated by an arrow A. As is explained while referring to FIG. 2, but not shown in FIG. 3, a reflection sheet, a diffusion sheet and a prism sheet are arranged on the backlight unit in FIG. 3.
Because of an optical need for the intensity of scattered light to be uniform across the entire light guide plate 122 that is used for the conventional backlight unit in FIG. 3, it may be necessary for the dot filling rate distribution, for example, to be continuously changed in the center area and in the four corner areas of the light guide plate 122. Therefore, a simple method has been studied that calls for the provision of an initial position to satisfy the obtained continuous distribution of the filling rate. For example, a method has been studied for forming a pattern wherein the filling rate is continuously changed by coupling areas having different lattice intervals. However, with this method, a defect encountered in many cases is that at the boundaries where filling rates are changed the seams are visible.
These seams can also be reduced by generating, across the entire plane on which dots are formed, a two-dimensional lattice whose shape is continuously changed to match the obtained filling rate distribution. However, excluding a case wherein the distribution is provided by a simple and easy analysis function, high level and extensive calculations are required for the generation of a lattice. That is, the conventional method used for calculating perturbations based on a lattice point is inadequate, not only for irregularities, but also for coping with the filling rate distribution.
For the above backlight unit, there is also a proposal according to which the structure is changed in order to improve the luminance or the angle of incidence of light. For example, a backlight unit wherein prisms are formed directly on the top of a light guide plate is proposed in xe2x80x9c""99 latest liquid crystal process techniquexe2x80x9d, Yoji Oki and Minoru Katsumata, Press Journal, Sep. 10, 2000, page 441. In addition, it is also proposed that an optical sheet, such as a diffusion sheet or a prism sheet, is not provided.
However, since the above proposals require that precise control be provided for the scattering mechanism of the light guide plate, the probability of occurrence of moire or an interference stripe will be increased, and stricter discussion of a method to be used for the reduction of moire is required. Furthermore, a discrete pattern, including a dot pattern having a higher quality, must be provided because the abovementioned conventional pseudo random dot patterns are inappropriate for coping with a high filling rate distribution, the uniformity of dot patterns is inappropriate, and depending on the structure of a backlight unit, the occurrence of a specific type of interference stripe can not be avoided. To provide this discrete pattern, a method is required whereby, in addition to the randomness of the discrete pattern, an index for a uniformity must be introduced to generate a random discrete pattern that satisfies a stricter condition.
Recently, to solve a problem wherein sample points are irregularly and uniformly extracted from a predetermined area in multi-dimensional space, the use of the LDS method has been discussed, especially for a mathematical field, such as numerical integration. For example, in xe2x80x9cIPSJ Magazinexe2x80x9d, Yoichi Ninomiya et al., Vol. 39, 1998, page 794, teaches that by using samples that are distributed irregularly and uniformly in an overall multi-dimensional space by using a determinative LDS, such as the Faure sequence or a Sobol sequence instead of pseudo random number, the price of a derivative can be quickly and accurately calculated by approximating a multi-dimensional integration solution as is done using the Monte Carlo method.
Further, as is explained in U.S. Pat. No. 5,872,725 and in Japanese Unexamined Patent Publication No. Hei 11-259452, the upper bound of discrepancy, which is a measure of a non-uniformity of point sets, is limited by an inequality in the above sequences. By using these sequences, the convergence of a multi-dimensional integration calculation, such as is used for the Monte Carlo method, can be quickly performed. Further, the use of numerical integration employing the LDS method is reported in order to increase the rendering speed attained by the ray tracing method.
As is described above, a pseudo random dot pattern that depends directly on randomness is inappropriate for an optical member that provides a preferable light guide plate, a backlight unit that uses the light guide plate, and a light-transmitting liquid crystal display device that uses this backlight unit. Therefore, a new method is required for controlling discrepancy and for generating an initial distribution. In this invention, a xe2x80x9cdiscrepancyxe2x80x9d is an index for the uniformity of the distribution of dispersed dots, as is described in xe2x80x9cDiscrepancy for pointsxe2x80x9d, for example, by Tezuka, in xe2x80x9cDiscrete structure and algorithm IVxe2x80x9d, edited by Kazuo Murota, Modern Science, Chapter 3.
To achieve the above objects, according to the present invention, an arrangement whereby the discrepancy is sufficiently low in a predetermined area is employed as the initial position for forming discrete patterns having improved randomness. As a result, uniform randomness is maintained for the discrete patterns. Further, in this invention, starting from a dot pattern with a low discrepancy as the initial pattern, the position of each dot is displaced by a repulsive force between dots having a definite size, so that the overlapping of the dots is eliminated.
In this invention, a process for removing overlapping dots, while assuming that a repulsive force is exerted between them, is defined as the repulsive force relaxation method. Even when a filling rate is high, the overlapping of dots is removed, using the repulsive force relaxation method, from an initial position with a low discrepancy, and a discrete pattern can be generated for which discrepancy remains low. In addition, since the filling rate and the discrepancy can be reduced for each predetermined area, even when the filling rate is continuously changed, a discrete pattern can be provided wherein satisfactory randomness is maintained and the filling rate is continuously changed, without causing a problem due to a change in the uniformity of the randomness of a dot pattern, i.e., a problem generated at a boundary whereat there is a change in the filling rate.
An optical member including the discrete pattern provided by this invention can uniformly provide a high luminance, without generating moire patterns.
The above objects of the invention are especially achieved by providing a discrete pattern according to the invention, an optical member that uses the discrete pattern, a light guide plate, a side light device, a light-transmitting liquid crystal display device, a discrete pattern generation method and a program that employs the discrete pattern generation method to generate discrete patterns, a computer-readable storage medium on which the program for generating discrete patterns is recorded, and a discrete pattern generation system.
According to the present invention, a discrete pattern, formed by dots discretely arranged in two dimensions, is provided wherein the dots included in a rectangular area having a longitudinal length of Lx and a transverse length of Ly satisfy expression (1),
Dxe2x89xa60.13Nxe2x88x921 15xe2x80x83xe2x80x83(1) 
(in expression (1), N denotes the number of dots included in a predetermined area, and D is obtained by expression (2), wherein A(x,y) defines the number of dots, of a total of N dots, included in a rectangular area for which a line segment extended from reference coordinates (0,0) to an arbitrary coordinate point (x,y) is a diagonal line),                               D          ⁡                      (                                          L                x                            ,                                                L                  y                                ;                N                                      )                          =                  ∫                                    ∫                                                L                  x                                ⁢                                  xL                  y                                                      ⁢                                                            [                                                                                    A                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    N                                        -                                          xy                                                                        L                          x                                                ⁢                                                  L                          y                                                                                                      ]                                2                            ⁢                                                                    ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                        L                    x                                    ⁢                                      L                    y                                                                                                          (        2        )            
and wherein S1 that is obtained by expression (3)                                           S            1                    ⁡                      (                                          r                1                            ,                              r                2                                      )                          =                              ∫                          r              1                                      r              2                                ⁢                                    ⅆ              r                        ⁢                          "LeftBracketingBar"                                                                    g                    1                                    ⁡                                      (                                                                  r                        ;                                                  r                          1                                                                    ,                                              r                        2                                                              )                                                  -                                  g                  av                                            "RightBracketingBar"                                                          (        3        )            
is equal to or smaller than 0.7. (In expression (3), g1 is obtained by dividing the average radial distribution function g(r) of each dot in the area by an integration value of g(r) over a range of from r1 to r2, and gav is the average value of g1 within the range of from r1 to r2. When the dots are arranged in a square lattice to satisfy a given filling rate, r1 and r2 are chosen as one and four times the value of the lattice constant Dr respectively. The dot filling rate is a value obtained by multiplying the square of the maximum diameter of a dot by the number of dots, and dividing the product by the size of the area.)
According to the invention, expression (1) is satisfied when the number of predetermined dots included in the area is equal to or smaller than 4000. D denotes the square of the discrepancy and for the dots included in the area, it is preferable that the exponent be smaller than xe2x88x921.15 at a predetermined filling rate.
It is preferable that S1 be equal to or smaller than 0.4, and that the discrete pattern be an arrangement wherein the average radial distribution function for the area is satisfactory smooth.
Each of the dots can have at least a two-dimensional or three-dimensional shape selected from a group including a polygon, a circle, a square, a rectangle, an ellipse, a circular conic and a polyhedron. The discrete pattern is so arranged that substantially adjacent dots are not overlapped. Further, the discrete pattern can be arranged without depending on the filling rate of the dots in the area. In addition, the discrete pattern can passively control a light beam. The control of the light beam is exercised by the scattering of light, the transmission of light or the absorption of light, and can be performed for a light guide plate, a light scattering plate, a dithering pattern, and a lithography photomask pattern.
According to the present invention, an optical member is provided on which a discrete pattern is formed by dots discretely arranged in two dimensions, wherein the dots included in a rectangular area having a longitudinal length of Lx and a transverse length of Ly satisfy expression (1),
Dxe2x89xa60.13Nxe2x88x921 15xe2x80x83xe2x80x83(1) 
(in expression (1), N denotes the number of dots included in a predetermined area, and D is obtained by expression (2), wherein A(x,y) defines the number of dots, of a total of N dots, included in a rectangular area for which a line segment extended from reference coordinates (0,0) to an arbitrary coordinate point (x,y) is a diagonal line),                               D          ⁡                      (                                          L                x                            ,                                                L                  y                                ;                N                                      )                          =                  ∫                                    ∫                                                L                  x                                ⁢                                  xL                  y                                                      ⁢                                                            [                                                                                    A                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    N                                        -                                          xy                                                                        L                          x                                                ⁢                                                  L                          y                                                                                                      ]                                2                            ⁢                                                                    ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                        L                    x                                    ⁢                                      L                    y                                                                                                          (        2        )            
and wherein S1 that is obtained by expression (3)                                           S            1                    ⁡                      (                                          r                1                            ,                              r                2                                      )                          =                              ∫                          r              1                                      r              2                                ⁢                                    ⅆ              r                        ⁢                          "LeftBracketingBar"                                                                    g                    1                                    ⁡                                      (                                                                  r                        ;                                                  r                          1                                                                    ,                                              r                        2                                                              )                                                  -                                  g                  av                                            "RightBracketingBar"                                                          (        3        )            
is equal to or smaller than 0.7. (In expression (3), g1 is obtained by dividing the average radial distribution function g(r) of each dot in the area by an integration value of g(r) over a range of from r1 to r2, and gav is the average value of g1 within the range of from r1 to r2. When the dots are arranged in a square lattice to satisfy a given filling rate, r1 and r2 are chosen as one and four times the value of the lattice constant Dr respectively. The dot filling rate is a value obtained by multiplying the square of the maximum diameter of a dot by the number of dots, and dividing the product by the size of the area.)
According to the invention, it is preferable that expression (1) be satisfied when the number of predetermined dots included in the area is equal to or smaller than 4000. D is the square of the discrepancy, and for the dots included in the area, the exponent is smaller than xe2x88x921.15 at a predetermined filling rate. S1 is equal to or smaller than 0.4, and that the discrete pattern be an arrangement wherein the average radial distribution function for the area is satisfactory smooth. Each of the dots has at least a two-dimensional or three-dimensional shape selected from a group including a polygon, a circle, a square, a rectangle, an ellipse, a circular conic and a polyhedron. The discrete pattern is so arranged that substantially adjacent dots are not overlapped. Further, the discrete pattern is arranged without depending on the filling rate of the dots in the area. In addition, the discrete pattern passively controls a light beam. The control of the light beam is exercised by the scattering of light, the transmission of light or the absorption of light, and is performed for a light guide plate, a light scattering plate, and a photomask.
According to the present invention, a light guide plate, used for a light-transmitting liquid crystal display device, is provided on which a discrete pattern is formed by dots discretely arranged in two dimensions, wherein the dots included in a rectangular area having a longitudinal length of Lx and a transverse length of Ly satisfy expression (1),
Dxe2x89xa60.13Nxe2x88x921 15xe2x80x83xe2x80x83(1) 
(in expression (1), N denotes the number of dots included in a predetermined area, and D is obtained by expression (2), wherein A(x,y) defines the number of dots, of a total of N dots, included in a rectangular area for which a line segment extended from reference coordinates (0,0) to an arbitrary coordinate point (x,y) is a diagonal line),                               D          ⁡                      (                                          L                x                            ,                                                L                  y                                ;                N                                      )                          =                  ∫                                    ∫                                                L                  x                                ⁢                                  xL                  y                                                      ⁢                                                            [                                                                                    A                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    N                                        -                                          xy                                                                        L                          x                                                ⁢                                                  L                          y                                                                                                      ]                                2                            ⁢                                                                    ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                        L                    x                                    ⁢                                      L                    y                                                                                                          (        2        )            
and wherein S1 that is obtained by expression (3)                                           S            1                    ⁡                      (                                          r                1                            ,                              r                2                                      )                          =                              ∫                          r              1                                      r              2                                ⁢                                    ⅆ              r                        ⁢                          "LeftBracketingBar"                                                                    g                    1                                    ⁡                                      (                                                                  r                        ;                                                  r                          1                                                                    ,                                              r                        2                                                              )                                                  -                                  g                  av                                            "RightBracketingBar"                                                          (        3        )            
is equal to or smaller than 0.7. (In expression (3), g1 is obtained by dividing the average radial distribution function g(r) of each dot in the area by an integration value of g(r) over a range of from r1 to r2, and gav is the average value of g1 within the range of from r1 to r2. When the dots are arranged in a square lattice to satisfy a given filling rate, r1 and r2 are chosen as one and four times the value of the lattice constant Dr respectively. The dot filling rate is a value obtained by multiplying the square of the maximum diameter of a dot by the number of dots, and dividing the product by the size of the area.)
According to the invention, expression (1) is satisfied when the number of predetermined dots included in the area is equal to or smaller than 4000. D is the square of the discrepancy, and for the dots included in the area, the exponent is smaller than xe2x88x921.15 at a predetermined filling rate. S1 is equal to or smaller than 0.4, and that the discrete pattern be an arrangement wherein the average radial distribution function for the area is satisfactory smooth. Each of the dots has at least a three-dimensional shape selected from a group including a polygon, a circle, a square, a rectangle, an ellipse, a circular conic and a polyhedron. The discrete pattern is so arranged that substantially adjacent dots are not overlapped. Further, the discrete pattern is arranged without depending on the filling rate of the dots in the area, and in corner areas of the light guide plate, the dots are arranged at a higher filling rate than that for the center area of the light guide plate. The light guide plate includes a display area wherein the discrete pattern is formed and a non-display area enclosing the display area.
According to the invention, a side light device comprises:
a light guide plate, on which a discrete pattern is formed by dots discretely arranged in two dimensions, wherein the dots included in a rectangular area having a longitudinal length of Lx and a transverse length of Ly satisfy expression (1),
Dxe2x89xa60.13Nxe2x88x921 15xe2x80x83xe2x80x83(1) 
(in expression (1), N denotes the number of dots included in a predetermined area, and D is obtained by expression (2), wherein A(x,y) defines the number of dots, of a total of N dots, included in a rectangular area for which a line segment extended from reference coordinates (0,0) to an arbitrary coordinate point (x,y) is a diagonal line),                               D          ⁡                      (                                          L                x                            ,                                                L                  y                                ;                N                                      )                          =                  ∫                                    ∫                                                L                  x                                ⁢                                  xL                  y                                                      ⁢                                                            [                                                                                    A                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    N                                        -                                          xy                                                                        L                          x                                                ⁢                                                  L                          y                                                                                                      ]                                2                            ⁢                                                                    ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                        L                    x                                    ⁢                                      L                    y                                                                                                          (        2        )            
and wherein S1 that is obtained by expression (3)                                           S            1                    ⁡                      (                                          r                1                            ,                              r                2                                      )                          =                              ∫                          r              1                                      r              2                                ⁢                                    ⅆ              r                        ⁢                          "LeftBracketingBar"                                                                    g                    1                                    ⁡                                      (                                                                  r                        ;                                                  r                          1                                                                    ,                                              r                        2                                                              )                                                  -                                  g                  av                                            "RightBracketingBar"                                                          (        3        )            
is equal to or smaller than 0.7 (In expression (3), g1 is obtained by dividing the average radial distribution function g(r) of each dot in the area by an integration value of g(r) over a range of from r1 to r2, and gav is the average value of g1 within the range of from r1 to r2. When the dots are arranged in a square lattice to satisfy a given filling rate, r1 and r2 are chosen as one and four times the value of the lattice constant Dr respectively. The dot filling rate is a value obtained by multiplying the square of the maximum diameter of a dot by the number of dots, and dividing the product by the size of the area.);
a light source for projecting light onto the light guide plate; and
a reflector for reflecting the light projected by the light source and transmitting the reflected light to the light guide plate.
According to the invention, expression (1) is satisfied when the number of predetermined dots included in the area is equal to or smaller than 4000. D is the square of the discrepancy, and for the dots included in the area, the exponent is smaller than xe2x88x921.15 at a predetermined filling rate. S1 is equal to or smaller than 0.4, and that the discrete pattern be an arrangement wherein the average radial distribution function for the area is satisfactory smooth. Each of the dots has at least a three-dimensional shape selected from a group including a polygon, a circle, an ellipse, a square, a rectangle, a circular conic and a polyhedron. The discrete pattern is so arranged that substantially adjacent dots are not overlapped. Further, the discrete pattern is arranged without depending on the filling rate of the dots in the area, and in corner areas of the light guide plate, the dots are arranged at a higher filling rate than that for the center area of the light guide plate. The light guide plate includes a display area wherein the discrete pattern is formed and a non-display area enclosing the display area.
Further, according to the invention, a light-transmitting liquid crystal display device comprises:
the above described side light device as a backlight unit.
According to the invention, a method for generating a discrete pattern wherein dots are discretely arranged in two dimensions comprises the steps of:
providing a predetermined area;
determining the number of dots to be arranged in the predetermined area; and
determined the position coordinates of the dots, so that the dots included in a rectangular area having a longitudinal length of Lx and a transverse length of Ly satisfy expression (1),
Dxe2x89xa60.13Nxe2x88x921 15xe2x80x83xe2x80x83(1) 
(in expression (1), N denotes the number of dots included in a predetermined area, and D is obtained by expression (2), wherein A(x,y) defines the number of dots, of a total of N dots, included in a rectangular area for which a line segment extended from reference coordinates (0,0) to an arbitrary coordinate point (x, y) is a diagonal line),                               D          ⁡                      (                                          L                x                            ,                                                L                  y                                ;                N                                      )                          =                  ∫                                    ∫                                                L                  x                                ⁢                                  xL                  y                                                            xe2x80x83                                      ⁢                                                            [                                                                                    A                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    N                                        -                                          xy                                                                        L                          x                                                ⁢                                                  L                          y                                                                                                      ]                                2                            ⁢                                                                    ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                        L                    x                                    ⁢                                      L                    y                                                                                                          (        2        )            
and wherein S1 that is obtained by expression (3)                                           S            1                    ⁡                      (                                          r                1                            ,                              r                2                                      )                          =                              ∫                          r              1                                      r              2                                ⁢                                    ⅆ              r                        ⁢                          "LeftBracketingBar"                                                                    g                    1                                    ⁡                                      (                                                                  r                        ;                                                  r                          1                                                                    ,                                              r                        2                                                              )                                                  -                                  g                  av                                            "RightBracketingBar"                                                          (        3        )            
is equal to or smaller than 0.7 (In expression (3), g1 is obtained by dividing the average radial distribution function g(r) of each dot in the area by an integration value of g(r) over a range of from r1 to r2, and gav is the average value of g1 within the range of from r1 to r2. When the dots are arranged in a square lattice to satisfy a given filling rate, r1 and r2 are chosen as one and four times the value of the lattice constant Dr respectively. The dot filling rate is a value obtained by multiplying the square of the maximum diameter of a dot by the number of dots, and dividing the product by the size of the area.);
setting the position coordinates as initial positions; and
changing the position coordinates of the dots so that the dots for which the position coordinates are determined do not overlap each other.
The step of determining the position coordinates includes the steps of:
generating and storing a first generator matrix for each coordinate axis;
employing the first generator matrix to generate and store a second generator matrix;
generating, as a first vector, the binary representation of a natural number n of a set of N natural numbers corresponding to N dots in the area;
generating a second vector using a product of the first vector and the second generator matrix;
generating the position coordinates of the dots while, for the coordinates, the elements of the second vector are defined as the values of the digits of a binary decimal number; and
increasing the natural number n by a predetermined number and generating position coordinates until the total number of repetitively generated point coordinates equals the number N of dots in the area. Further, the discrete pattern generation method further comprises the step of: generating a discrete pattern having a predetermined size by employing a predetermined boundary condition between any two of multiple areas. The step of changing the position coordinates includes the step of calculating a repulsive force between adjacent dots depending on their distances and their sizes. The step of changing the position coordinates includes the steps of:
calculating a repulsive force exerted by another dot located near a predetermined dot;
displacing the predetermined dot in accordance with the magnitude of the repulsive force;
calculating a repulsive force for the predetermined dot that is displaced; and
calculating a total for the repulsive forces of dots located within a predetermined range.
The discrete pattern generation method further comprises the step of: repetitively performing the step of changing the position coordinates until a predetermined convergence condition is established by a difference between a total of first potential energy and a total of second potential energy, which are calculated immediately before getting the total of first potential energy. When the interval between the dots is equal to or smaller than a predetermined value, the repulsive force is substantially constant, and when the interval exceeds the predetermined value, the repulsive force is reduced in accordance with the increase in the interval. The initial positions of the dots are obtained by using low-discrepancy sequences.
According to the invention, a program is provided for executing a method for generating a discrete pattern wherein dots are discretely arranged in two dimensions, the program comprising the steps of:
providing a predetermined area;
determining the number of dots to be arranged in the predetermined area; and
determined the position coordinates of the dots, so that the dots included in a rectangular area having a longitudinal length of Lx and a transverse length of Ly satisfy expression (1),
Dxe2x89xa60.13Nxe2x88x921 15xe2x80x83xe2x80x83(1) 
(in expression (1), N denotes the number of dots included in a predetermined area, and D is obtained by expression (2), wherein A(x,y) defines the number of dots, of a total of N dots, included in a rectangular area for which a line segment extended from reference coordinates (0,0) to an arbitrary coordinate point (x, y) is a diagonal line),                               D          ⁡                      (                                          L                x                            ,                                                L                  y                                ;                N                                      )                          =                  ∫                                    ∫                                                L                  x                                ⁢                                  xL                  y                                                            xe2x80x83                                      ⁢                                                            [                                                                                    A                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    N                                        -                                          xy                                                                        L                          x                                                ⁢                                                  L                          y                                                                                                      ]                                2                            ⁢                                                                    ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                        L                    x                                    ⁢                                      L                    y                                                                                                          (        2        )            
and wherein S1 that is obtained by expression (3)                                           S            1                    ⁡                      (                                          r                1                            ,                              r                2                                      )                          =                              ∫                          r              1                                      r              2                                ⁢                                    ⅆ              r                        ⁢                          "LeftBracketingBar"                                                                    g                    1                                    ⁡                                      (                                                                  r                        ;                                                  r                          1                                                                    ,                                              r                        2                                                              )                                                  -                                  g                  av                                            "RightBracketingBar"                                                          (        3        )            
is equal to or smaller than 0.7 (In expression (3), g1 is obtained by dividing the average radial distribution function g(r) of each dot in the area by an integration value of g(r) over a range of from r1 to r2, and gav is the average value of g1 within the range of from r1 to r2. When the dots are arranged in a square lattice to satisfy a given filling rate, r1 and r2 are chosen as one and four times the value of the lattice constant Dr respectively. The dot filling rate is a value obtained by multiplying the square of the maximum diameter of a dot by the number of dots, and dividing the product by the size of the area.);
setting the position coordinates as initial positions; and
changing the position coordinates of the dots so that the dots for which the position coordinates are determined do not overlap each other.
The step of determining the position coordinates includes the steps of:
generating and storing a first generator matrix for each coordinate axis;
employing the first generator matrix to generate and store a second generator matrix,
generating, as a first vector, the binary representation of a natural number n of a set of N natural numbers corresponding to N dots in the area;
generating a second vector using a product of the first vector and the second generator matrix;
generating the position coordinates of the dots while, for the coordinates, the elements of the second vector are defined as the values of the digits of a binary decimal number; and
increasing the natural number n by a predetermined number and generating position coordinates until the total number of repetitively generated point coordinates equals the number N of dots in the area. Further, the program further comprises the step of generating a discrete pattern having a predetermined size by employing a predetermined boundary condition between any two of multiple areas. The step of changing the position coordinates includes the step of calculating a repulsive force between adjacent dots depending on their distances and their sizes. The step of changing the position coordinates includes the steps of:
calculating a repulsive force exerted by another dot located near a predetermined dot;
displacing the predetermined dot in accordance with the magnitude of the repulsive force;
calculating a repulsive force for the predetermined dot that is displaced; and
calculating a total for the repulsive forces of dots located within a predetermined range. The program further comprises the step of: repetitively performing the step of changing the position coordinates until a predetermined convergence condition is established by a difference between a total of first potential energy and a total of second potential energy, which are calculated immediately before getting the total of first potential energy. When the interval between the dots is equal to or smaller than a predetermined value, the repulsive force is substantially constant, and when the interval exceeds the predetermined value, the repulsive force is reduced in accordance with the increase in the interval. It is preferable that the initial positions of the dots be obtained by using low-discrepancy sequences.
According to the invention, a computer-readable storage medium is provided on which a program is stored that executes a method for generating a discrete pattern wherein dots are discretely arranged in two dimensions, the program comprising the steps of:
providing a predetermined area;
determining the number of dots to be arranged in the predetermined area; and
determined the position coordinates of the dots, so that the dots included in a rectangular area having a longitudinal length of Lx and a transverse length of Ly satisfy expression (1),
Dxe2x89xa60.13Nxe2x88x921 15xe2x80x83xe2x80x83(1) 
(in expression (1), N denotes the number of dots included in a predetermined area, and D is obtained by expression (2), wherein A(x,y) defines the number of dots, of a total of N dots, included in a rectangular area for which a line segment extended from reference coordinates (0,0) to an arbitrary coordinate point (x, y) is a diagonal line),                               D          ⁡                      (                                          L                x                            ,                                                L                  y                                ;                N                                      )                          =                  ∫                                    ∫                                                L                  x                                ⁢                                  xL                  y                                                            xe2x80x83                                      ⁢                                                            [                                                                                    A                        ⁡                                                  (                                                      x                            ,                            y                                                    )                                                                    N                                        -                                          xy                                                                        L                          x                                                ⁢                                                  L                          y                                                                                                      ]                                2                            ⁢                                                                    ⅆ                    x                                    ⁢                                      ⅆ                    y                                                                                        L                    x                                    ⁢                                      L                    y                                                                                                          (        2        )            
and wherein S1 that is obtained by expression (3)                                           S            1                    ⁡                      (                                          r                1                            ,                              r                2                                      )                          =                              ∫                          r              1                                      r              2                                ⁢                                    ⅆ              r                        ⁢                          "LeftBracketingBar"                                                                    g                    1                                    ⁡                                      (                                                                  r                        ;                                                  r                          1                                                                    ,                                              r                        2                                                              )                                                  -                                  g                  av                                            "RightBracketingBar"                                                          (        3        )            
is equal to or smaller than 0.7 (In expression (3), g1 is obtained by dividing the average radial distribution function g(r) of each dot in the area by an integration value of g(r) over a range of from r1 to r2, and gav is the average value of g1 within the range of from r1 to r2. When the dots are arranged in a square lattice to satisfy a given filling rate, r1 and r2 are chosen as one and four times the value of the lattice constant Dr respectively. The dot filling rate is a value obtained by multiplying the square of the maximum diameter of a dot by the number of dots, and dividing the product by the size of the area.);
setting the position coordinates as initial positions; and
changing the position coordinates of the dots so that the dots for which the position coordinates are determined do not overlap each other.
The step of determining the position coordinates includes the steps of:
generating and storing a first generator matrix for each coordinate axis;
employing the first generator matrix to generate and store a second generator matrix;
generating, as a first vector, the binary representation of a natural number n of a set of N natural numbers corresponding to N dots in the area;
generating a second vector using a product of the first vector and the second generator matrix;
generating the position coordinates of the dots while, for the coordinates, the elements of the second vector are defined as the values of the digits of a binary decimal number; and
increasing the natural number n by a predetermined number and generating position coordinates until the total number of repetitively generated point coordinates equals the number N of dots in the area. Further, the program further comprises the step of: generating a discrete pattern having a predetermined size by employing a predetermined boundary condition between any two of multiple areas. The step of changing the position coordinates includes the step of calculating a repulsive force between adjacent dots depending on their distances and their sizes. The step of changing the position coordinates includes the steps of:
calculating a repulsive force exerted by another dot located near a predetermined dot;
displacing the predetermined dot in accordance with the magnitude of the repulsive force;
calculating a repulsive force for the predetermined dot that is displaced; and
calculating a total for the repulsive forces of dots located within a predetermined range. The program further comprises the step of: repetitively performing the step of changing the position coordinates until a predetermined convergence condition is established by a difference between a total of first potential energy and a total of second potential energy, which are calculated immediately before getting the total of first potential energy. When the interval between the dots is equal to or smaller than a predetermined value, the repulsive force is substantially constant, and when the interval exceeds the predetermined value, the repulsive force is reduced in accordance with the increase in the interval. It is preferable that the initial positions of the dots be obtained by using low-discrepancy sequences.
According to the invention, a discrete pattern generation system, for generating the above described discrete pattern, comprises:
means for providing the discrete pattern;
storage means for storing the position coordinates of the dots that form the discrete pattern;
printer means for outputting the position coordinates included in the recording means; and
pattern receiving elements wherein the discrete pattern is formed by the printer means.
Various other objects, features, and attendant advantages of the present invention will become more fully appreciated as the same becomes better understood when considered in conjunction with the accompanying drawings, in which like reference characters designate the same or similar parts throughout the several views.